Measurement and simulation strategies for characterizing the response of a given system to an input, e.g., the electrical behavior of interconnect and packaging structures for electronic circuits, often utilize a frequency-domain approach. Physically, such an approach is equivalent to applying sinusoidal excitations to the terminals, or ports, of a structure and then measuring the response at the same and/or other ports. The relation between excitations and responses, and how those relations vary with sinusoidal frequency, are used to calculate scattering parameters. For a structure with p ports, measured at K frequencies (where K is typically hundreds or thousands), the scattering-(or S-) parameter data is a set of K matrices, where each matrix is a set of p×p complex numbers. The entry in the ith row and jth column of the matrix associated with a particular frequency f indicates how a sinusoidal excitation with frequency f applied to port j will affect the response at port i.
Circuit simulators, which are typically used to compute the time evolution of voltages and currents in complicated circuits, often require accurate representations of the electrical behavior of interconnect and packaging used to couple circuit blocks. For this reason, almost all commercial circuit simulators have some method for converting models of packaging and interconnect represented using frequency-domain S-parameter data into models that are suitable for time-domain simulation. A wide variety of methods are in common use, with convolution-based approaches being the most established.
The more modern and now preferred strategy for using S-parameter data in time-domain circuit simulation is to construct a p-input, p-output system of linear differential equations whose response to sinusoid excitations closely matches the responses represented by the corresponding S-parameter data. Such a system of differential equations, usually referred to as a state-space model, is easily included in time-domain circuit simulation.
The typical form for an nth-order state-space model isE{dot over (x)}(t)=Ax(t)+Bu(t)  (1)y(t)=Cx(t)+Du(t)  (2)where x(t) is an n-length vector and represents the model's state, u(t) is a p-length vector representing the input to the model, y(t) is a p-length vector representing the output of the model, and A, E, B, C, and D are n×n, n×n, n×p, p×n, and p×p real matrices, respectively. A is the “state transition” matrix that specifies the dynamics of the modeled system. One challenge in extracting a state-space model from S-parameter data is determining the order of the model, n, as well as the entries of the matrices A, E, B, C, and D, such that the frequency response (i.e., the response to sinusoids of different frequencies) of the state-space model matches the S-parameter data.
The response of a linear state-space model to a sinusoidal input is a sinusoid of the same frequency, so the frequency response of a state-space system is easily determined analytically. The variable s is commonly used to represent a generalized frequency of excitation, leading to the identity
      ⅆ          ⅆ      t        →      s    .  This identity is then used to determine the frequency response of a state-space model, as inH(s)=C(sE−A)−1B+D.  (3)where H(s) is a p×p matrix of complex values that depend on s. For an accurate state-space model, when s is equal to the purely imaginary value s=j2πf (where j≡√{square root over (−1)}), H(s) should match the given S-parameter data at frequency f.
Almost all of the state-of-the-art algorithms for extracting state-space models from S-parameter data are based on a general multi-step fitting-based strategy:
1) The order of the model, n, is selected via iterative rational fitting.
2) The state-space system poles or resonant modes (equivalently, the matrices A and E) are determined by a frequency-response matching procedure. The fitting procedure is most commonly applied to each of p single-input p-output “columns” of relations, and the resulting sets of poles are combined.
3) The unstable poles are removed, as such poles correspond to models whose behavior will grow unrealistically towards infinity.
4) With A and E now fixed, the B, C, and D matrices are selected so as to match the S-parameter data as closely as possible.
5) If desired, the values of B, C, and D are perturbed so as to ensure the state-space model is passive (does not generate energy). For systems having no source of external power, e.g., packaging and interconnect schemes, violating passivity generates conspicuous errors.
While generally useful, these iterative approaches to rational fitting are fundamentally uncontrollable, and therefore not robust. There is no mathematical proof that iterative rational fitting converges, and there are many examples where such methods generate unstable state-space models.
There are many alternatives for each step of the above five-step procedure. However, existing methods for extracting a state-space model based on projection or matching approaches are unable to determine the state-space model directly from S-parameter data. Existing projection strategies inherently sample only a fraction of the data, leading to models with unreliable accuracy, and existing matching approaches can be numerically unstable. In addition, both projection and matching methods typically generate unstable models, particularly when the S-parameter data is based on actual measurement data.